On the Spectrum of the Discrete Schrödinger Operator of a Rank-Two Perturbation on \(\mathbb{Z}\)

We consider a family of Schrödinger operators \(\widehat{H}_{\lambda\mu k}=-\Delta-\lambda\delta_{k,\cdot}-\mu\delta_{0,\cdot}\) on the one-dimensional lattice \(\mathbb{Z}\) , where \(\Delta\) is a standard discrete Laplacian, \(\delta_{\cdot,\cdot}\) is a Kronecker delta function, and \(\lambda,\mu\in\mathbb{R}\) and \(k\in\mathbb{Z}\) are parameters. Eigenvalue behavior of the operators and their dependence on the parameters are explicitly derived. Moreover, we obtain asymptotics for the eigenvalues as the distance between two elements of the potential function support approaches infinity.